Certain observations on antisymmetric \(T_0\)-quasi-metric spaces (Q2077298)

Let \((X,q)\) be a quasi-metric space. \begin{itemize} \item[(a)] A pair \((x, y) \in X \times X\) is called {\em antisymmetric} pair if it satisfies the condition \(q(x, y) \neq q(y,x)\). For any \(x \in X\), observe that \((x,x)\) is not antisymmetric. \item[(b)] A finite sequence \((x_i)_{i=0}^n\) of points in \(X\) such that \(x=x_0,x_1,x_2,\dots,x_{n-1},x_n=y\) is called a {\em (finite) antisymmetric path} provided that all the pairs \((x_i, x_{i+1})\) are antisymmetric whenever \(i \in \{0,\dots,n-1\}\). \end{itemize} The set \(R_q\) of antisymmetric pairs of the space \((X,q)\) is defined by \[R_q=\{(x,y) \in X \times X: q(x,y)\neq q(y,x)\}.\] The complement \(Z_q\) of \(R_q\) is the set of symmetric pairs of \((X, q)\). A quasi-metric space \((X,q)\) is called {\em antisymmetric} if \[Z_q = \{(x, x): x \in X\} = \Delta_X,\] where \(\Delta_X\) denotes the diagonal of \(X\). In other words, \(q\) is an antisymmetric quasi-metric on \(X\) if and only if \[R_q = X \times X \setminus \Delta_X = \{(x, y): x \neq y\}.\] This article gives a successful answer to the question: when does a topological space \(X\) admit an antisymmetric quasi-metric inducing its topology? Some answers to this question are achieved by imposing a few conditions on the topological space.